How many numbers between $1$ and $100$ (inclusive) are divisible by $5$ or $3$ ?
Explanation: There are $20$ numbers divisible by $5$ between $1$ and $100$, and $33$ numbers divisible by $3$ between $1$ and $100$. So, you might think there are $20 + 33 = 53$ numbers divisible by one or the other, but this is overcounting something. We're counting every number which is divisible by both $5$ and $3$ twice. So, for example, $15$ is counted once as a number divisible by $5$, and then again as a number divisible by $3$. So, we need to count how many numbers are divisible by both $5$ and $3$ and subtract this from what we had before. Being divisible by both $5$ and $3$ is the same thing as being divisible by $15$, so there are $6$ numbers between $1$ and $100$ divisible by both. Subtracting, there are $53 - 6 = 47$ numbers divisible by $5$ or $3$.